# What is the Maclaurin series of #f(x) = cos(x)#?

##### 1 Answer

Nov 1, 2015

A Maclaurin series is simply the Taylor series centered around

#sum_(n=1)^(N) (f^((n))(0))/(n!) x^n#

#= (f(0))/(0!)x^0 + (f'(0))/(1!)x^1 +(f''(0))/(2!)x^2 + (f'''(0))/(3!)x^3 + ...#

Thus, we need to take derivatives until we see a unique pattern.

#f^((0))(x) = color(green)(f(x) = cosx)#

#color(green)(f'(x) = -sinx)#

#color(green)(f''(x) = -cosx)#

#color(green)(f'''(x) = sinx)#

#color(green)(f''''(x) = cosx)#

So stopping at

#= (cos(0))/(0!)x^0 + cancel((-sin(0))/(1!)x^1)^(0) +(-cos(0))/(2!)x^2 + cancel((sin(0))/(3!)x^3)^(0) + ...#

#= 1/(0!) - x^2/(2!) + x^4/(4!) - x^6/(6!) + ...#

#color(blue)(= 1 - x^2/2 + x^4/(24) - x^6/(720) + ...)#